Solve the exponential equation for $x$. 5 6 x + 5 125 3 x − 1 = 5 − 5 x + 2 \dfrac{5\^{ 6x+5}}{125\^{ 3x-1}}=5\^{ -5x+2} $x=$
Solution: The strategy Let's write $125$ in base $5$. Then, using the properties of exponents, we can express the entire left hand side of the equation as $5$ raised to some linear function. Finally, we can equate the exponents of the resulting equation to solve for the unknown. Simplifying the left hand side 5 6 x + 5 125 3 x − 1 = 5 6 x + 5 ( 5 3 ) 3 x − 1 = 5 6 x + 5 5 9 x − 3 = 5 6 x + 5 − ( 9 x − 3 ) = 5 − 3 x + 8 ( 125 = 5 3 ) ( ( a n ) m = a n ⋅ m ) ( a n a m = a n − m ) \begin{aligned}\dfrac{5\^{ 6x+5}}{125\^{ 3x-1}}&=\dfrac{5\^{ 6x+5}}{(5^3)\^{ 3x-1}}&&&&(125=5^3) \\\\\\\\ &=\dfrac{5\^{ C{6x+5}}}{5\^{ {9x-3}}} &&&&((a^n)^m=a^{n\cdot m})\\\\\\\\ &=5\^{ C{6x+5} \ - \ ({9x-3})}&&&&(\dfrac{a^n}{a^m}=a^{n-m})\\\\\\\\ &=5\^{ -3x+8} \end{aligned} Solving the equation We obtain the following equation. 5 − 3 x + 8 = 5 − 5 x + 2 5\^{-3x+8}=5\^{ -5x+2} Now we can equate the exponents and solve for $x$. $\begin{aligned} -3x+8 &=-5x+2\\\\ x &=-3\end{aligned}$ The answer The answer is $x=-3$. You can check this answer by substituting $\it{x=-3}$ in the original equation and evaluating both sides.